Multiplier algebra

The multiplier algebra , based on the English term also called multiplier algebra , is a concept from the mathematical theory of C * algebras . It is about the maximum embedding of a C * -algebra as an essential two-sided ideal in a C * -algebra with one element.

Definitions

Centralizers

Let it be a C * -algebra. A linear mapping is called a left or right centralizer , if any ${\displaystyle A}$ ${\displaystyle \rho :A\rightarrow A}$

${\displaystyle \rho (ab)=\rho (a)b}$or for everyone . ${\displaystyle \rho (ab)=a\rho (b)}$ ${\displaystyle a,b\in A}$

A double centralizer is a pair where ${\displaystyle (\rho _{1},\rho _{2})}$

• ${\displaystyle \rho _{1}}$ is a legal centralizer,
• ${\displaystyle \rho _{2}}$ is a left centralizer and
• ${\displaystyle \rho _{1}(a)b=a\rho _{2}(b)}$for everyone . ${\displaystyle a,b\in A}$

Multipliers

We now introduce the concept of the multiplier, the definition of which requires a Hilbert space . We will then relate the term multiplier to the centralizers in order to ensure independence from the choice of Hilbert space.

According to Gelfand-Neumark's theorem, a C * -algebra can be understood without restriction as a C * -subalgebra of the operator algebra of bounded , linear operators on a Hilbert space , so that only applies to for all . One then says, operate on in a non-degenerate manner . An operator is called a left or right multiplier , if and . A bilateral multiplier, or simply multiplier, is an operator that is both left and right multiplier. ${\displaystyle A}$ ${\displaystyle B(H)}$ ${\displaystyle H}$ ${\displaystyle a\xi =0}$ ${\displaystyle a\in A}$ ${\displaystyle \xi =0}$ ${\displaystyle A}$ ${\displaystyle H}$ ${\displaystyle x\in B(H)}$ ${\displaystyle xA\subset A}$ ${\displaystyle Ax\subset A}$ ${\displaystyle B(H)}$

If there is a left or right multiplier, then a left or right centralizer is given by or obviously. If a multiplier is, then it is a centralizer. One can show that in this situation the mappings are bijective functions of the set of all left, right or both-sided multipliers on the set of all left, right or double centralizers. In particular, the multiplier terms do not depend on the choice of Hilbert space on which non-degenerate operates. ${\displaystyle x}$ ${\displaystyle \rho _{x}(a):=xa}$ ${\displaystyle {\tilde {\rho }}_{x}(a):=ax}$ ${\displaystyle x}$ ${\displaystyle ({\tilde {\rho }}_{x},\rho _{x})}$ ${\displaystyle x\mapsto \rho _{x},x\mapsto {\tilde {\rho }}_{x},x\mapsto ({\tilde {\rho }}_{x},\rho _{x})}$ ${\displaystyle A}$

The set of all multipliers is obviously a C * -algebra, it is called the multiplier algebra of . By design, a two-sided ideal is in . is even an essential ideal in , that is, has an average different from 0 with each two-sided ideal different from 0. ${\displaystyle M(A)}$ ${\displaystyle A}$ ${\displaystyle A}$ ${\displaystyle M(A)}$ ${\displaystyle A}$ ${\displaystyle M(A)}$ ${\displaystyle A}$

Strict topology

In addition to the standard topology, the so-called strict topology is also considered on the multiplier algebra . This is the locally convex topology that is generated by all semi-norms . ${\displaystyle M(A)}$ ${\displaystyle p_{a}(x):=\|ax\|+\|xa\|,a\in A}$

Examples

• If a unit has 1, then it is , because then applies to every left multiplier . ${\displaystyle A}$ ${\displaystyle M(A)=A}$ ${\displaystyle x}$ ${\displaystyle x=x1\in A}$
• If the C * -algebra of compact operators is over a Hilbert space , then is . ${\displaystyle K}$ ${\displaystyle H}$ ${\displaystyle M(K)=B(H)}$
• Let be the commutative C * -algebra of the C ₀ -functions on a locally compact Hausdorff space . Then is isomorphic to the C * -algebra of the bounded, continuous functions on and this is again isomorphic to the C * -algebra of the continuous functions on the Stone-Čech compactification . It is well known that Stone-Čech compacting is, according to its universal property, a "greatest" compacting. In the case of general C * algebras, the following generalization of this topological state of affairs applies: ${\displaystyle A=C_{0}(X)}$ ${\displaystyle X}$ ${\displaystyle M(A)}$ ${\displaystyle X}$ ${\displaystyle C(\beta X)}$ ${\displaystyle \beta X}$

• If a C * -algebra, which is a two-sided, essential ideal in a C * -algebra , then there is an injective * -homomorphism whose restriction is to the identity . ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle B\rightarrow M(A)}$ ${\displaystyle A}$

The transition to multiplier algebra can therefore be described as "non-commutative Stone-Čech compactification".

• If with a locally compact Hausdorff space and a C * -algebra , then is isomorphic to the C * -algebra of all continuous functions , whereby the strict topology carries. ${\displaystyle A=C_{0}(X)\otimes B}$ ${\displaystyle X}$ ${\displaystyle B}$ ${\displaystyle M(A)}$ ${\displaystyle \beta X\rightarrow M(B)}$ ${\displaystyle M(B)}$

More terms

If a C * -algebra is called the outer algebra. The outer algebra of the C * algebra of compact operators is the Calkin algebra . ${\displaystyle A}$ ${\displaystyle Q(A):=M(A)/A}$ ${\displaystyle K}$

Since the multiplier algebra of a C * -algebra with a unit element does not bring anything new, you first tensor with to get to a C * -algebra without a unit element, and then form the multiplier algebra or outer algebra: ${\displaystyle K}$

${\displaystyle M^{s}(A):=M(A\otimes K)}$, . ${\displaystyle Q^{s}(A):=Q(A\otimes K)}$

This is called the stable multiplier algebra or stable outer algebra. Stability plays an important role in the K-theory of C * algebras. The following applies:

• For every C * -algebra is and , where 0 denotes the trivial one-element group . In short: the K groups of a stable multiplier algebra disappear. ${\displaystyle A}$ ${\displaystyle K_{0}(M^{s}(A))=0}$ ${\displaystyle K_{1}(M^{s}(A))=0}$

As an application we show

${\displaystyle K_{i}(A)\cong K_{1-i}(Q^{s}(A)),\,i=0,1}$

To do this, we consider the one from the short exact sequence

${\displaystyle 0\rightarrow A\otimes K\rightarrow M^{s}(A)\rightarrow Q^{s}(A)\rightarrow 0}$

Cyclic exact sequence obtained using Bott periodicity

${\displaystyle {\begin{array}{ccccc}K_{0}(A\otimes K)&\rightarrow &K_{0}(M^{s}(A))&\rightarrow &K_{0}(Q^{s}(A))\\\uparrow &&&&\downarrow \\K_{1}(Q^{s}(A))&\leftarrow &K_{1}(M^{s}(A))&\leftarrow &K_{1}(A\otimes K)\\\end{array}}}$

Since the middle groups of each line disappear according to the above sentence, the vertical arrows must be isomorphisms for the sake of accuracy. Since the K-theory is invariant to stabilization, that is, it holds , the above claim follows. ${\displaystyle K_{i}(A)\cong K_{i}(A\otimes K),\,i=0,1}$

Individual evidence

1. ↑ Gert K. Pedersen: C * -Algebras and Their Automorphism Groups , Academic Press Inc. (1979), ISBN 0-1254-9450-5 , Section 3.12.1
2. ^ Gert K. Pedersen: C * -Algebras and Their Automorphism Groups , Academic Press Inc. (1979), ISBN 0-1254-9450-5 , Theorem 3.12.3
3. ^ Gert K. Pedersen: C * -Algebras and Their Automorphism Groups , Academic Press Inc. (1979), ISBN 0-1254-9450-5 , sentence 3.12.8
4. ↑ C. Akemann, G. Pedersen, J Tomiyama: Multipliers of C * -algebras , Jornal of Functional Analysis, Volume 13 (1973), pages 277-301
5. ↑ Bruce Blackadar: K-Theory for Operator Algebras , Springer Verlag (1986), ISBN 3-540-96391-X , Theorem 12.2.1
6. ↑ Bruce Blackadar: K-Theory for Operator Algebras , Springer Verlag (1986), ISBN 3-540-96391-X , Corollary 12.2.3