Strong Morita Equivalence

Idea

For rings, the classical Morita theorem, which says that for Morita equivalent rings $R,S$ there is an $R-S$-bimodule ${}_R X_S$ that takes ${}_S M$ to ${}_R(X\otimes_S M)$. For $C^\ast$-algebras, if one uses the Hilbert space representations as the category of modules, there is no such theorem.

Given a right Hilbert $B$-module $X_B$, a left action of $A$ by adjointable operators converts representations $\pi$ of $B$ to representations $X-\operatorname{Ind}\pi$ of $A$ on $X\otimes_{B} \mathcal{H}_\pi$. Rieffel’s notion of strong Morita equivalence extends this to a equivalence relation on $C^\ast$-algebras implemented by a suitable Hilbert module.

Definition

Definition(Imprimitivity bimodule) . Let $A,B$ be $C^\ast$-algebras. An $A-B$-imprimitivity bimodule is an full $A-B$-Hilbert bimodule such that for $x,y\in X$, $a\in A$, $b\in B$, $$ \begin{gather} \langle a\cdot x,y \rangle_B = \langle x, a^\ast\cdot y\rangle_B\\\ {}_A\langle x\cdot b,y\rangle = {}_A\langle x, y\cdot b^\ast \rangle\\\ {}_A\langle x,y\rangle\cdot z = x\cdot\langle y,z\rangle_B \end{gather} $$

Some points to be made:

  1. The conditions say that $A$ acts by adjointable operators on $X_{B}$ and $B$ acts by adjointable operators on ${}_{A}X$ and two algebra-valued inner products are compatible.
  2. A Hilbert $A$-module $X$ is full if the ideal $I=\operatorname{span}\{\langle x,y\rangle_A | x,y\in X\}$ is dense in $A$. The fullness makes left and right multiplications adjointable automatically, so what is really needed is the third condition.
  3. The adjoints of a operator on a Hilbert module does not automatically exist. Every adjointable map $rho:X\to Y$ between Hilbert $A$-modues is a bounded linear $A$-module map.
Definition(Strong Morita equivalence) . A pair of $C^\ast$ algebras $A,B$ are strongly Morita equivalence if there is an $A-B$-imprimitivity bimodule ${}_A X_B$.
Remark(Correspondence) . An imprimitivity bimodule is a special kind of correspondence: $(A,B,X_B,\phi:A\to\mathcal{L}(X))$. The situation is analogues to that an anaequivalence is a special kind of anafunctor.

Internal Tensor Product and Equivalence

The internal tensor product ${}_A X_B\otimes {}_B Y_C$, of two imprimitivity bimodules $X,Y$, is defined in such a way that it is in turn an $A-C$-imprimitivity bimodule, thus providing transitivity for strong Morita equivalence.

Definition(Internal tensor product) . If ${}_A X_B,{}_B Y_C$ are imprimitivity bimodules, then the internal tensor product $X\otimes_B Y$ is the completion of the $B$-balanced algebraic tensor product $X\odot_{B} Y$ with respect to the $A$- and $C$-valued pre-inner products $$ \begin{gather} \langle x\otimes_B y, z\otimes_B w\rangle_C = \langle \langle z,x\rangle_B\cdot y, w\rangle_C\\\ {}_A\langle x\otimes_B y, z\otimes_B w\rangle = {}_A\langle x,z\cdot {}_B\langle w \cdot\ y\rangle\rangle. \end{gather} $$

$B$-balanced means $(x\cdot b) \otimes_B y = x\otimes_B (b\cdot y)$, that is, $X\odot_B Y = X\odot Y / \sim$ where $\sim$ is the subspace spanned by $\{(x\cdot b)\otimes y - x\otimes(b\cdot y)| x\in X, y\in Y, b\in B\}$.

The internal tensor product is in turn an $A-C$-imprimitivity bimodule.

Remark. The definition of internal tensor product is conventient while restrictive. For two Hilbert modules $X_A,{}_B Y$ together with a $\ast$-homomorphism $\phi: A\to \mathcal{L}(B)$, one can quotient out the subspace spanned by $\{(x\cdot a)\otimes y - y\otimes\phi(a)x\}$ to form an algebraic tensor product $(X\odot_A Y)_B$ that is a right $B$-module. Define a $B$-valued pre-inner product on it via $$ \langle x_1\otimes y_1,x_2\otimes y_2\rangle_B = \langle y_1,\phi(\langle x_1,x_2\rangle_A) y_2\rangle_B $$ and take the completion. This is a general definition for the internal tensor product. For imprimitivity bimodules the homomorphisms $\phi$ are given implicitly.

Some Results

  1. Two unital $C^\ast$-algebras are Strongly Morita equivalent iff they are Morita equivalent as abstract algebras.
  2. Two $\sigma$-unital $C^\ast$-algebras (have countable approximate identity) are strongly Morita equivalent if they are stably isomorphic.
  3. Two separable $C^\ast$-algebras (the spectrums are separable, i.e. have countable dense subset) are strongly Morita equivalent iff they are stably isomorphic.

There is another characterzation of Morita equivalence by corners. A corner is a $C^\ast$-subalgebra of the form $pAp$ where $p$ is a self-adjoit idempotent in the multiplier algebra of $A$. If $ApA$ is dense in $A$ then it is a full corner. The opposite corner to $pAp$ is $(1-p)A(1-p)$, now

Theorem(Brown-Green-Rieffel) . Let $A,B$ be $C^\ast$-algebras, they are strongly Morita equivalent iff they both embed as opposite full corners of another $C^\ast$-algebra $C$.

The bicategory $\mathsf{C}^\ast-\operatorname{Alg}$ formed by $C^\ast$-algebras with morphisms imprimitivity bimodules and 2-morphisms intertwiners form a $(2,1)$-category, i.e., all the intertwiners are isomorphisms.