K-theory of Banach algebras

The K-theory of Banach algebras is a concept from the mathematical field of functional analysis . It provides invariants for Banach algebras , which are algebras investigated in functional analysis that generalize some known function spaces and operator algebras such as spaces of continuous or integrable functions or algebras of continuous linear operators on Banach spaces based on essential common properties.

It generalizes the topological K-theory , which deals with the study of vector bundles on topological spaces , to general Banach algebras, with the C * algebras playing an important role. The topological K-theory of compact spaces can be reformulated as the K-theory of the Banach algebras of continuous functions and then transferred to any Banach algebras, even the one element of the algebras can be dispensed with. Since the assignment is a contravariant functor from the category of compact Hausdorff spaces to the category of Banach algebras and since the topological K-theory is also contravariant, we get a covariant functor from the category of Banach algebras to the category of Abelian groups . ${\displaystyle X}$ ${\displaystyle C(X)}$ ${\displaystyle X\rightarrow \mathbb {C} }$ ${\displaystyle X\mapsto C(X)}$

Since non-commutative algebras can also occur here, one speaks of non-commutative topology. The K-theory is an important subject of investigation in the theory of C * algebras. Below is a -Banachalgebra, go out by adjunction of the element produced. ${\displaystyle A}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle A^{+}}$ ${\displaystyle A}$

K ₀ of Banach algebras

The vector bundles of topological K-theory correspond to the algebraic side of the finally generated , projective modules and these are direct summands in free modules , can do so through idempotents a sufficiently large matrix algebra over are described. For the idempotents there are different, suitable equivalence terms , which all coincide if one goes into the inductive limit , where equivalent idempotents belong to stable-isomorphic, projective modules. One possible definition is that two idempotents are called idempotents and equivalent if there is one such that and elements exist with . The equivalence class of 'll be using designated. If one has two idempotents and , then one can replace with an equivalent idempotent , so that , then there is again an idempotent. If one sets , a well-defined semigroup connection is given on the set of equivalence classes from idempotents . From this one could again form the associated Grothendieck group , but a small technical change is made to the definition of the group in order to be able to adequately handle algebras without a single element, for example ideals in Banach algebras. It is defined as a subgroup of the Grothendieck group of , namely as the set of all differences , where are idempotent such that . ${\displaystyle A^{n}}$ ${\displaystyle p\in M_{n}(A)}$ ${\displaystyle A}$ ${\displaystyle M_{\infty }(A)=\mathrm {ind} _{n\to \infty }M_{n}(A)}$ ${\displaystyle p}$ ${\displaystyle q}$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle p,q\in M_{n}(A)}$ ${\displaystyle x,y\in M_{n}(A)}$ ${\displaystyle p=xy,q=yx}$ ${\displaystyle p}$ ${\displaystyle [p]}$ ${\displaystyle p}$ ${\displaystyle q}$ ${\displaystyle q}$ ${\displaystyle q'}$ ${\displaystyle pq'=0}$ ${\displaystyle p+q'}$ ${\displaystyle [p]+[q]:=[p+q']}$ ${\displaystyle V(A)}$ ${\displaystyle M_{\infty }(A)}$ ${\displaystyle K_{0}(A)}$ ${\displaystyle K_{0}(A)}$ ${\displaystyle V(A^{+})}$ ${\displaystyle [p]-[q]}$ ${\displaystyle p,q\in M_{\infty }(A^{+})}$ ${\displaystyle p-q\in M_{\infty }(A)}$

If a two-sided , closed ideal is obtained from the short, exact sequence ${\displaystyle J\subset A}$

${\displaystyle 0\rightarrow J\rightarrow A\rightarrow A/J\rightarrow 0}$

an exact sequence

${\displaystyle K_{0}(J)\rightarrow K_{0}(A)\rightarrow K_{0}(A/J)}$,

which in general cannot be continued exactly with 0, neither to the left nor to the right.

The definition is laid out in such a way that applies to compact spaces ( Serre and Swan's theorem ). In the case of C * algebras, one can replace the idempotents with orthogonal projections , that is, with self-adjoint idempotents , and get the same result, since every idempotent is equivalent to a projection. Can be made using K as an important application of ₀ , the AF C * -algebras classify. ${\displaystyle K_{0}(C(X))=K^{0}(X)}$ ${\displaystyle X}$

K ₁ from Banach algebras

For the definition of we define the set of all invertible matrices from whose image in the quotient algebra equal to the identity matrix is. Means ${\displaystyle K_{1}(A)}$ ${\displaystyle GL_{n}(A)}$ ${\displaystyle M_{n}(A^{+})}$ ${\displaystyle M_{n}(A^{+})/M_{n}(A)\cong M_{n}(\mathbb {C} )}$

${\displaystyle GL_{n}(A)\ni x\mapsto {\begin{pmatrix}x&0\\0&1\end{pmatrix}}\in GL_{n+1}(A)}$

We understand it as a subgroup of and provide the resulting inductive limes with the final topology . The connected component of the unit is a normal divisor and one defines ${\displaystyle GL_{n}(A)}$ ${\displaystyle GL_{n+1}(A)}$ ${\displaystyle GL_{\infty }(A):=\mathrm {ind} _{n\to \infty }GL_{n}(A)}$ ${\displaystyle GL_{\infty }(A)_{0}}$

${\displaystyle K_{1}(A):=GL_{\infty }(A)/GL_{\infty }(A)_{0}=\mathrm {ind} _{n\to \infty }GL_{n}(A)/GL_{n}(A)_{0}}$.

Despite the non-commutativity of the matrix algebras, the group so defined turns out to be commutative. While in the algebraic K-theory the commutator group is separated out to define the K ₁ group (see Abelization ), in the topological K-theory for Banach algebras the connected component of the unit is used. In the case of C * algebras, the invertible elements can be replaced by unitary elements in the above construction and the same result is obtained. ${\displaystyle K_{1}(A)}$

If a two-sided, closed ideal is obtained from the short, exact sequence ${\displaystyle J\subset A}$

${\displaystyle 0\rightarrow J\rightarrow A\rightarrow A/J\rightarrow 0}$

an exact sequence

${\displaystyle K_{1}(J)\rightarrow K_{1}(A)\rightarrow K_{1}(A/J)}$,

which in general cannot be continued exactly with 0, neither to the left nor to the right.

Again, the definition is laid out in such a way that applies to compact rooms . If one denotes with the Banach algebra of all continuous functions that vanish in infinity , provided with the supremum norm , one can show. One calls the suspension of ; it is the Banachach algebra version of the suspension or reduced suspension of topological spaces. Iteration of the suspension could be used to define higher K groups, for example , but this is not necessary because of the Bott periodicity , which is also valid here . ${\displaystyle K_{1}(C(X))=K^{1}(X)}$ ${\displaystyle X}$ ${\displaystyle SA}$ ${\displaystyle \mathbb {R} \rightarrow A}$ ${\displaystyle K_{1}(A)\cong K_{0}(SA)}$ ${\displaystyle SA}$ ${\displaystyle A}$ ${\displaystyle K_{n}(A):=K(S^{n}A)}$

Cyclic sequence

As in the topological K-theory, an index mapping and a Bott isomorphism can be constructed so that the above exact sequences combine to form the following cyclic exact sequence:

${\displaystyle {\begin{array}{ccccc}K_{0}(J)&\rightarrow &K_{0}(A)&\rightarrow &K_{0}(A/J)\\\uparrow &&&&\downarrow \\K_{1}(A/J)&\leftarrow &K_{1}(A)&\leftarrow &K_{1}(J)\\\end{array}}}$

This sequence is very useful when calculating K groups. If some groups of the sequence are known, this allows conclusions to be drawn about those that are still unknown due to the accuracy.

Other properties

Functoriality

Let it be a continuous homomorphism between Banach algebras. This defines homomorphisms that are compatible with the above constructions of the K groups and thus lead to group homomorphisms and . This makes and covariant functors between the category of Banach algebras and the category of Abelian groups. ${\displaystyle \varphi :A\rightarrow B}$ ${\displaystyle \varphi _{n}:M_{n}(A)\rightarrow M_{n}(B)}$ ${\displaystyle K_{0}(\varphi ):K_{0}(A)\rightarrow K_{0}(B)}$ ${\displaystyle K_{1}(\varphi ):K_{1}(A)\rightarrow K_{1}(B)}$ ${\displaystyle K_{0}}$ ${\displaystyle K_{1}}$

Homotopy invariance

Two continuous homomorphisms between Banach algebras are called homotopic if there is a family of homomorphisms such that each is continuous and holds. Homotopic homomorphisms induce the same group homomorphisms between the K groups. ${\displaystyle \varphi ,\psi :A\rightarrow B}$ ${\displaystyle (\varphi _{t})_{t\in [0,1]}}$ ${\displaystyle t\mapsto \varphi _{t}(a)}$ ${\displaystyle a\in A}$ ${\displaystyle \varphi _{0}=\varphi ,\varphi _{1}=\psi }$

stability

If a Banach algebra, then applies to and all . If there is an inductive limit in the category of Banach algebras, then applies ${\displaystyle A}$ ${\displaystyle K_{i}(M_{n}(A))\cong K_{i}(A)}$ ${\displaystyle i=0,1}$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle A=\mathrm {ind} _{j\in J}A_{j}}$

${\displaystyle K_{i}(\mathrm {ind} _{j\in J}A_{j})\cong \mathrm {ind} _{j\in J}K_{i}(A_{j}),\quad i=0,1}$.

The compatibility with the formation of the inductive limit results directly from the constructions of the K groups by means of inductive limits.

Especially for C * -algebras and the inductive limit of the category of C * -algebras is isomorphic to the tensor product , where the C * -algebra of the compact operators is over a separable Hilbert space . This applies to . ${\displaystyle M_{n}(A)\cong M_{n}(\mathbb {C} )\otimes A}$ ${\displaystyle M_{n}(A)}$ ${\displaystyle A\otimes K}$ ${\displaystyle K}$ ${\displaystyle K_{i}(A\otimes K)\cong K_{i}(A)}$ ${\displaystyle i=0,1}$

literature

• Jacek Brodzki: An Introduction to K-theory and Cyclic Cohomology. arxiv : funct-an / 9606001 .
• Bruce Blackadar: K-Theory for Operator Algebras. Springer Verlag, 1986, ISBN 3-540-96391-X .

Web links

• Christian Voigt: K-theory of operator algebras. (PDF).

swell

1. ↑ Blackadar : K-Theory for Operator Algebras. Springer Verlag, 1986, ISBN 3-540-96391-X .