--- a/src/LazyTensors/lazy_tensor_operations.jl	Sun Oct 18 22:21:43 2020 +0200
+++ b/src/LazyTensors/lazy_tensor_operations.jl	Sun Oct 18 22:30:17 2020 +0200
@@ -141,3 +141,24 @@
 function apply_transpose(llm::LazyLinearMap{T,R,D}, v::AbstractArray{T,R}, I::Vararg{Index,D}) where {T,R,D}
     apply(LazyLinearMap(llm.A, llm.domain_indicies, llm.range_indicies), v, I...)
 end
+
+
+"""
+    LazyIdentity{T,D} <: TensorMapping{T,D,D}
+
+The lazy identity TensorMapping for a given size. Usefull for building up higher dimensional tensor mappings from lower
+dimensional ones through outer products. Also used in the Implementation for InflatedTensorMapping.
+"""
+struct LazyIdentity{T,D} <: TensorMapping{T,D,D}
+    size::NTuple{D,Int}
+end
+export LazyIdentity
+
+LazyIdentity{T}(size::NTuple{D,Int}) where {T,D} = LazyIdentity{T,D}(size)
+
+range_size(tmi::LazyIdentity) = tmi.size
+domain_size(tmi::LazyIdentity) = tmi.size
+
+apply(tmi::LazyIdentity{T,D}, v::AbstractArray{T,D}, I::Vararg{Index,D}) where {T,D} = v[Int.(I)...]
+apply_transpose(tmi::LazyIdentity{T,D}, v::AbstractArray{T,D}, I::Vararg{Index,D}) where {T,D} = v[Int.(I)...]
+

--- a/test/testLazyTensors.jl	Sun Oct 18 22:21:43 2020 +0200
+++ b/test/testLazyTensors.jl	Sun Oct 18 22:30:17 2020 +0200
@@ -259,4 +259,28 @@
 
 end
 
+
+@testset "LazyIdentity" begin
+    @test LazyIdentity{Float64}((4,5)) isa LazyIdentity{T,2} where T
+    @test LazyIdentity{Float64}((4,5)) isa TensorMapping{T,2,2} where T
+    A = rand(3,4)
+    Ã = LazyLinearMap(A, (1,), (2,))
+    v = rand(4)
+
+    for sz ∈ [(4,5),(3,),(5,6,4)]
+        I = LazyIdentity{Float64}(sz)
+        v = rand(sz...)
+        @test I*v == v
+        @test I'*v == v
+
+        @test range_size(I) == sz
+        @test domain_size(I) == sz
+    end
+
+    I = LazyIdentity{Float64}((4,5))
+    v = rand(4,5)
+    @inferred (I*v)[3,2]
+    @inferred range_size(I)
 end
+
+end