It consists of those infinite square matrices with entries in the hyperfinite type II 1 factor that define bounded operators. From Wikipedia, the free encyclopedia. Categories : Von Neumann algebras.

## Uniqueness of the Injective III1 Factor | SpringerLink

Namespaces Article Talk. For all common algebraic structures, and, in particular for vector spaces , an injective homomorphism is also called a monomorphism. However, in the more general context of category theory , the definition of a monomorphism differs from that of an injective homomorphism.

A function f that is not injective is sometimes called many-to-one. Let f be a function whose domain is a set X.

This principle is referred to as the horizontal line test. Functions with left inverses are always injections.

## On the uniqueness of injective III$_1$ factor

In this case, g is called a retraction of f. Conversely, f is called a section of g.

- Hyperfinite type II factor.
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In other words, an injective function can be "reversed" by a left inverses, but is not necessarily invertible , which requires that the function is bijective. More generally, injective partial functions are called partial bijections.

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A proof that a function f is injective depends on how the function is presented and what properties the function holds. For functions that are given by some formula there is a basic idea.

### Acta Mathematica

Advertisement Hide. Pages Connes' reduction of the uniqueness proof to the bicentralizer problem.

Haagerup's solution of the bicentralizer problem.